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Here's one you won't find on Kayak. They are cheapest on the days they fly to a particular city. Their 2 main bases are Las Vegas and Phoenix-Mesa. Their planes take off from one of the bases taking passengers back home. The plane is on the ground for 50 minutes and goes back to base with a new load.
If you just have carry on you can avoid fees and waiting for the baggage cart. http://www.allegiantair.com/

Sprit and Allegiant are ones that most follow the European low-fare model. Fares of Southwest, Jetblue, Virgin America, Frontier and AirTran are usually matched by the "legacy" carriers, so I don't consider them low-cost.

Of course, Allegiant has only 52 planes and Spirit 35, and they only serve some of the markets in the US. You should first tell which route you're flying.

As other people have mentioned, I have never found airlines like Southwest or JetBlue particularly cheap.

If travelling from Europe, I would try to use a travel agency specialising in cheap airfares, there are lots of them.

Get them to work out a price for you, based on multiple flights. In the past I've used airline and online agencies like expedia and have plugged in multi-city options, and I've always got back an outrageous fare.

A couple of years ago I flew Munich - NY - Chicago - Vegas - DC in April for 600 Euros including taxes, the alternative pricing per the airlines website was around 1800 euros

Allegiant flies from Youngstown and Huntington West Virginia to Myrtle Beach, SC. If the schedule works for you, rent a car in Myrtle Beach and drive north to Cape Hatteras. Do you have a week or less?
Amtrak goes to Norfolk (with bus to VA Beach) from Washington DC. Fly to Reagan National, take the Metro to Union Station to catch your train.